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## Numerical tests

In this section several numerical tests are performed using the derived above numerical stochastic schemes. In some tests we consider also a spatial scheme of Osher (details are given in [Toro1999]) and a kinetic scheme for one-dimensional pressureless gas taken from [BouchutJin2003]. These two schemes satisfy the entropy increase condition. The scheme, which satisfy the entropy increase condition, will be called classical scheme. In order to minimize the numerical diffusion, we set the Courant-Friedrichs-Lewy constant close to one, but still meeting the eq. (3.58): 𝑐𝑓𝑙 = 0.99. Backward and direct step, ramp, triangle and hat initial velocity profiles in one-dimensional case are considered for Riemann problem. Averaged numerical solutions are compared with analytical solutions (if the latter exist) or numerical solution of corresponding PDF equation. Further, the SPDEs (3.42)-(3.43) are solved for the considered initial profiles of velocity, and solutions are compared to numerical solution of the transport equation (3.16) for the velocity PDF.

Following abbreviations are used in the legends of figures:

∙ Stochastic Godunov method (SGod)

∙ Stochastic Lax-Friedrichs method (SL-F)

∙ Stochastic Lax-Wendroff method (SL-W)

∙ Stochastic GForce method (SGF)

∙ Exact solution, used as reference (Ex sol)

A logarithmic scale is used for spatial and statistical convergence.

### Computational domain and grid

In order to perform the LES calculation, we have to extend the 2D domain used for the RANS calculation (described in section 6.2.1) into a three-dimensional one. Given the computational cost of the resolution of the lateral boundary layers, we add only a width of 2ℎ in the third direction 𝑍. B. Sainte-Rose has shown that due to the use of periodic boundary conditions at the walls 𝑍 ≡ 𝑐𝑜𝑛𝑠𝑡 the choice of computational domain depth does not influence the solution [SainteRose2010]. In the 𝑍 direction the grid is uniform and consists in 30 cells, thus Δ𝑍 is about 0.0023 m. As for 2D case, the grid is refined in the 𝑋 and 𝑌 directions at the corner of the step, in the recirculation region and in near the lower wall of the chamber. At the corner of the step a grid resolution Δ𝑋 is equal to Δ𝑌 and is 1 × 10−4 m. Figure 6.39 illustrates a slice of the 3D grid at 𝑍 = ℎ. The total number of inner cells is 4 106 010.

#### Computational strategy

The physical domain was divided into 256 parts distributed on 256 Nehalem nodes consisting of two 2.80 GHz quad-core processors. In order to obtain time-averaged fields for one considered model starting from a solution calculated by another model, around 210 hours were necessary.

After a transition time which corresponds to 3 residence times in the combustion chamber (6.1), unsteady fields are averaged. The averaged time for the LES in all four cases is approximately 0.21 s, that corresponds to 11 residence times in the chamber. The timeaveraged vertical profiles at 10 residence times are compared with the time-averaged profiles at 11 residence times at 𝑋 points where the experimental data at 𝑍 = ℎ are available (see, e.g. fig. 6.46). The obtained numerical profiles of ⟨ ̃︀ 𝑇⟩, ⟨̃︀𝑢𝑥⟩, ⟨̃︀𝑢𝑦⟩, ⟨𝑢𝑅𝑀𝑆 𝑥 ⟩, ⟨𝑢𝑅𝑀𝑆 𝑦 ⟩ at 10

characteristic times superimpose visually to the respective profiles at 11 characteristic times.

**Table of contents :**

Contents

List of symbols

List of dimensionless numbers

List of abbreviations

**1 Introduction **

2 Background

2.1 Turbulence characteristics

2.1.1 Homogeneous isotropic turbulence

The Kolmogorov hypothesis

2.2 Flame structure

2.2.1 Premixed flame Regimes in premixed combustion

2.2.2 Non-premixed flame

2.3 Navier-Stokes equations for aerothermochemistry

2.3.1 Conservation of mass and species

2.3.2 Chemical kinetics

2.3.3 Conservation of momentum

2.3.4 Conservation of total energy

2.3.5 State law of an ideal gas

2.4 Reynolds averaged Navier-Stokes (RANS) approach

2.4.1 Definition of the ensemble average

2.4.2 Averaged Navier-Stokes equations

2.4.3 Closure of the RANS equations

2.4.4 Turbulence models

2.5 Large eddy simulation (LES)

2.5.1 Closure of the LES equations

2.6 General models of turbulent combustion

2.6.1 PDF approach

Eulerian PDFs

Presumed PDF

Transported Eulerian PDF

2.6.2 Partially stirred reactor models

Eddy dissipation concept

Partially stirred reactor model

Extended partially stirred reactor model

Unsteady partially stirred reactor model

2.6.3 Thickened flame model

**3 Eulerian (Field) Monte Carlo methods for solving the Favre one-tim one-point velocity PDF transport equation **

3.1 Description of the problem

3.1.1 One-dimensional model PDF equation

3.1.2 One-dimensional SPDEs

3.1.3 Equivalence between SPDEs and PDF equation

3.2 Stochastic numerical schemes

3.2.1 Schemes for partial differential equations

Numerical notations for stochastic schemes

Numerical stochastic schemes

Mean density conservation in stochastic schemes

3.2.2 Schemes for stochastic partial differential equations General remarks

3.3 Numerical tests

3.3.1 Backward step velocity profile Numerical solution

3.3.2 Scheme non-dissipativity test

3.3.3 Statistically homogeneous velocity fluctuations test

3.3.4 Model PDF equation with non-zero RHS

PDF

SPDEs

Numerical solution

3.4 Conclusions

3.5 Perspectives

**4 Flamelet tabulated chemistry (FTC) model **

4.1 Original implementation of FTC beta-PDF model in CEDRE

4.1.1 Modeling of Favre-averaged progress variable

4.1.2 Modeling of Favre variance of progress variable

Transport equation

Algebraic expression

4.1.3 Modeling of Favre-averaged mixture fraction

4.1.4 Modeling of Favre variance of mixture fraction

Transport equation

Algebraic expression

4.1.5 Gradient of Favre-averaged progress variable

4.1.6 Coupling between tabulated chemistry and CEDRE

4.1.7 Presumed beta-PDF

4.1.8 Algorithm of beta-PDF integration

4.2 Updated FTC beta-PDF model

4.2.1 New models for dissipation and source terms

4.2.2 Gradient of Favre-averaged progress variable

4.2.3 Algorithm of beta-PDF integration

4.2.4 Numerical implementation of beta-PDF integration Study of convergence of the semi-analytic beta integration method

**5 Transported partially stirred reactor (TPaSR) model **

5.1 Original EPaSR model 151

5.2 TPaSR model: EPaSR adaptation to CEDRE

6 Backward-facing step flow

6.1 A3C experimental setup

6.2 RANS nonreactive backward-facing step flow calculation

6.2.1 Computational domain and grid

6.2.2 Physical models

6.2.3 Boundary conditions

6.2.4 Numerical methods

6.2.5 Computational strategy

6.2.6 Results

Recirculation region

Favre-averaged velocity

RMS velocity fluctuations

Turbulent frequency

6.2.7 Conclusions

6.3 RANS reactive backward-facing step flow calculation

6.3.1 Numerical setup

6.3.2 Combustion modeling

6.3.3 Results

Quasi-laminar approaches

PFTC 𝛽-PDF models

TPaSR model

6.3.4 Conclusions

6.4 LES reactive backward-facing step flow calculation

6.4.1 Computational domain and grid

6.4.2 Physical models

6.4.3 Boundary conditions

6.4.4 Combustion modeling and numerical schemes

6.4.5 Computational strategy

6.4.6 Results

Mean temperature

Recirculation region

Mean streamwise velocity

Mean transverse velocity

Mean mass fractions

Velocity fluctuations

Impact of reflecting boundary conditions

6.4.7 Conclusions

6.5 Conclusions

**7 Conclusions and future work **

7.1 Conclusion

7.2 Future work

7.2.1 SPDEs

7.2.2 TPaSR model

7.2.3 FTC with presumed beta-PDF model

**A Some aspects of SPDEs modelling **

A.1 Random choice method (RCM) for solution satisfying entropy increase condition

A.2 Stochastic Runge-Kutta method

A.3 Schemes for SPDEs statistically equivalent to the joint velocity-scalar PDF equation

A.4 𝑛 dimensional space schemes for PDEs

A.5 Numerical tests

A.5.1 Step velocity profile

PDF

PDEs

Numerical solution

A.5.2 Ramp velocity profile

PDF

PDEs

Numerical solution

A.5.3 Triangle velocity profile

PDF

PDEs

Numerical olution

A.5.4 Hat velocity profile

PDEs

Numerical solution

A.5.5 Scheme non-dissipativity test

Numerical solution

A.6 Study of Langevin equation

**B Dependence of solution on computational grid in reactive backwardfacing step flow **

B.1 Numerical setup

B.2 Results

B.2.1 RANS/PFTC beta-PDF

B.2.2 RANS/TPaSR

List of figures

List of tables

**Bibliography**